Complete Set of Representatives Problems

At this point in our course we have covered Induction, Euclid's Algorithm, Unique Factorization, Congruence, Congruence Classes, and Rings/Fields.

I have tried solving this problem several ways. I know that the set is a complete representatives if the set consists of 11 integers (which it does) and no integer in the set is congruent to any other integer in the set. So, i have set a^k=(congruent)b^k (mod 11) where a,b E Set with a>b, and then tried to use this to figure for which values of k this can be true (and then by finding that all other k make the set a complete set of representatives) but I cannot seem to find a way to solve for k.

At this point in our course we have covered Induction, Euclid's Algorithm, Unique Factorization, Congruence, Congruence Classes, and Rings/Fields.

I have tried solving this problem several ways. I know that the set is a complete representatives if the set consists of 11 integers (which it does) and no integer in the set is congruent to any other integer in the set. So, i have set a^k=(congruent)b^k (mod 11) where a,b E Set with a>b, and then tried to use this to figure for which values of k this can be true (and then by finding that all other k make the set a complete set of representatives) but I cannot seem to find a way to solve for k.

Any help would be very much appriciated! Thanks!

I am not sure what the question is asking. Are you asking for which $\displaystyle k\in\mathbb{N}$ is $\displaystyle f:\mathbb{Z}_{11}\to\mathbb{Z}_{11}:z\mapsto z^k$ a bijection?